Simplify the following using identities.
a.\[{\left( {109} \right)^2} + {\left( {91} \right)^2}\]
b.\[{\left( {200} \right)^2} - {\left( {100} \right)^2}\]
c.\[{\left( {1025} \right)^2} - {\left( {975} \right)^2}\]
d.\[{\left( {10} \right)^2} + {\left( {20} \right)^2}\]

Hint: Firstly, observe the question and then use the identity \[{\left( {a + b} \right)^2} + {\left( {a - b} \right)^2} = 2\left[ {{a^2} + {b^2}} \right]\] , \[{a^2} - {b^2} = \left( {a - b} \right)\left( {a + b} \right)\] , \[{\left( {a + b} \right)^2} - {\left( {a - b} \right)^2} = 4ab\] , \[{a^2} + {b^2} = {\left( {a + b} \right)^2} - 2ab\] . With the formula we can easily simplify this equation by knowing all the variables i.e a and b.

Formula Used: Here, we can use the formula according to the requirement of the question\[{\left( {a + b} \right)^2} + {\left( {a - b} \right)^2} = 2\left[ {{a^2} + {b^2}} \right]\] , \[{a^2} - {b^2} = \left( {a - b} \right)\left( {a + b} \right)\] , \[{\left( {a + b} \right)^2} - {\left( {a - b} \right)^2} = 4ab\] , \[{a^2} + {b^2} = {\left( {a + b} \right)^2} - 2ab\]

Complete step-by-step answer:
Now, we will begin with:
(a)\[{\left( {109} \right)^2} + {\left( {91} \right)^2}\]
First, split 109 and 91 in factors of 100.
\[ \Rightarrow {\left( {100 + 9} \right)^2} + {\left( {100 - 9} \right)^2}\]
Here, it becomes an identity \[{\left( {a + b} \right)^2} + {\left( {a - b} \right)^2} = 2\left[ {{a^2} + {b^2}} \right]\]
Now considering left hand side , \[{\left( {a + b} \right)^2} + {\left( {a - b} \right)^2} = \]\[{\left( {100 + 9} \right)^2} + {\left( {100 - 9} \right)^2}\]
From here, we can clearly see that \[a = 100\] and \[b = 9\] .
Put the values of a and b in right hand side of formula \[ \Rightarrow 2\left[ {{a^2} + {b^2}} \right]\]
\[ \Rightarrow 2\left[ {{{\left( {100} \right)}^2} + {{\left( 9 \right)}^2}} \right]\]
By opening the squares:
 \[ \Rightarrow 2\left[ {10000 + 81} \right]\]
On further simplifying:
\[ \Rightarrow 2\left[ {10081} \right]\]
We get, \[ \Rightarrow 20162\]
\[{\left( {109} \right)^2} + {\left( {91} \right)^2}\]\[ \Rightarrow 20162\]

(b)\[{\left( {200} \right)^2} - {\left( {100} \right)^2}\]
As, it is clearly visible that is \[{a^2} - {b^2}\] and hence we can use the identity \[{a^2} - {b^2} = \left( {a - b} \right)\left( {a + b} \right)\] .
Now considering left hand side , \[{a^2} - {b^2}\]\[ = {\left( {200} \right)^2} - {\left( {100} \right)^2}\]
From here, we can clearly see that \[a = 200\] and \[b = 100\] .
Put the values of a and b in right hand side of formula \[ \Rightarrow \left( {a - b} \right)\left( {a + b} \right)\]
\[ \Rightarrow \left( {200 - 100} \right)\left( {200 + 100} \right)\]
On simplifying:
\[ \Rightarrow 100 * 300\]
We get, \[ \Rightarrow 30000\]
\[{\left( {200} \right)^2} - {\left( {100} \right)^2}\]\[ \Rightarrow 30000\]

(c)\[{\left( {1025} \right)^2} - {\left( {975} \right)^2}\]
First, of all split 1025 and 975 in factors of 1000.
\[ \Rightarrow {\left( {1000 + 25} \right)^2} - {\left( {1000 - 25} \right)^2}\]
Here, it becomes an identity \[{\left( {a + b} \right)^2} - {\left( {a - b} \right)^2} = 4ab\]
Now considering left hand side , \[{\left( {a + b} \right)^2} - {\left( {a - b} \right)^2} = \]\[{\left( {1000 + 25} \right)^2} + {\left( {1000 - 25} \right)^2}\]
From here, we can clearly see that \[a = 1000\] and \[b = 25\] .
Put the values of a and b in right hand side of formula \[ \Rightarrow 4ab\]
\[ \Rightarrow 4*1000*25\]
By multiplying:
\[ \Rightarrow 100000\]
\[{\left( {1025} \right)^2} - {\left( {975} \right)^2}\]\[ \Rightarrow 100000\]

(d)\[{\left( {10} \right)^2} + {\left( {20} \right)^2}\]
As, it is clearly visible that is \[{a^2} + {b^2}\] and hence we can use the identity \[{a^2} + {b^2} = {\left( {a + b} \right)^2} - 2ab\]
Now considering left hand side , \[{a^2} + {b^2}\]\[ = {\left( {10} \right)^2} + {\left( {20} \right)^2}\]
From here, we can clearly see that \[a = 10\] and \[b = 20\] .
Put the values of a and b in right hand side of formula \[ = {\left( {a + b} \right)^2} - 2ab\]
\[ \Rightarrow {\left( {10 + 20} \right)^2} - 2 * 10 * 20\]
\[ \Rightarrow {\left( {30} \right)^2} - 2 * 10 * 20\]
On simplifying square,
\[ \Rightarrow 900 - 2 * 10 * 20\]
And on multiplying,
\[ \Rightarrow 900 - 400\]
We get, \[ \Rightarrow 500\]
\[{\left( {10} \right)^2} + {\left( {20} \right)^2}\]\[ \Rightarrow 500\]

Note: In this type of question, first of all observe the given statements and calculate the values. Accordingly, implement the identities which are most appropriate.